On asymptotic properties of matrix semigroups with an invariant cone

Recently, several research efforts showed that the analysis of joint spectral characteristics of sets of matrices can be simplified when these matrices share an invariant cone. We prove new results in this direction.We prove that the joint spectral subradius is continuous in the neighborhood of sets of matrices that leave an embedded pair of cones invariant.We show that both the averaged maximal spectral radius, as well as the maximal trace, where the maximum is taken on all the products of the same length t, converge towards the joint spectral radius when t increases, provided that the matrices share an invariant cone, and additionally one of them is primitive.